Theorem dvacos 33497

Description: Derivative of arccosine. (Contributed by Brendan Leahy, 18-Dec-2018.)

Hypothesis

Ref	Expression
dvasin.d	|- D = ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) )

Assertion

Ref	Expression
dvacos	|- ( CC _D (arccos |` D ) ) = ( x e. D |-> ( -u 1 / ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) )

Distinct variable group:   x, D

Proof of Theorem dvacos

Step	Hyp	Ref	Expression
1		df-acos 24593	. . . . 5 |- arccos = ( x e. CC |-> ( ( pi / 2 ) - (arcsin ` x ) ) )
2	1	reseq1i 5392	. . . 4 |- (arccos |` D ) = ( ( x e. CC |-> ( ( pi / 2 ) - (arcsin ` x ) ) ) |` D )
3		dvasin.d	. . . . . 6 |- D = ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) )
4		difss 3737	. . . . . 6 |- ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) C_ CC
5	3, 4	eqsstri 3635	. . . . 5 |- D C_ CC
6		resmpt 5449	. . . . 5 |- ( D C_ CC -> ( ( x e. CC |-> ( ( pi / 2 ) - (arcsin ` x ) ) ) |` D ) = ( x e. D |-> ( ( pi / 2 ) - (arcsin ` x ) ) ) )
7	5, 6	ax-mp 5	. . . 4 |- ( ( x e. CC |-> ( ( pi / 2 ) - (arcsin ` x ) ) ) |` D ) = ( x e. D |-> ( ( pi / 2 ) - (arcsin ` x ) ) )
8	2, 7	eqtri 2644	. . 3 |- (arccos |` D ) = ( x e. D |-> ( ( pi / 2 ) - (arcsin ` x ) ) )
9	8	oveq2i 6661	. 2 |- ( CC _D (arccos |` D ) ) = ( CC _D ( x e. D |-> ( ( pi / 2 ) - (arcsin ` x ) ) ) )
10		cnelprrecn 10029	. . . . 5 |- CC e. { RR , CC }
11	10	a1i 11	. . . 4 |- ( T. -> CC e. { RR , CC } )
12		halfpire 24216	. . . . . 6 |- ( pi / 2 ) e. RR
13	12	recni 10052	. . . . 5 |- ( pi / 2 ) e. CC
14	13	a1i 11	. . . 4 |- ( ( T. /\ x e. D ) -> ( pi / 2 ) e. CC )
15		c0ex 10034	. . . . 5 |- 0 e. _V
16	15	a1i 11	. . . 4 |- ( ( T. /\ x e. D ) -> 0 e. _V )
17	13	a1i 11	. . . . 5 |- ( ( T. /\ x e. CC ) -> ( pi / 2 ) e. CC )
18	15	a1i 11	. . . . 5 |- ( ( T. /\ x e. CC ) -> 0 e. _V )
19	13	a1i 11	. . . . . 6 |- ( T. -> ( pi / 2 ) e. CC )
20	11, 19	dvmptc 23721	. . . . 5 |- ( T. -> ( CC _D ( x e. CC |-> ( pi / 2 ) ) ) = ( x e. CC |-> 0 ) )
21	5	a1i 11	. . . . 5 |- ( T. -> D C_ CC )
22		eqid 2622	. . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
23	22	cnfldtopon 22586	. . . . . . 7 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
24	23	toponunii 20721	. . . . . . . 8 |- CC = U. ( TopOpen ` ℂfld )
25	24	restid 16094	. . . . . . 7 |- ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
26	23, 25	ax-mp 5	. . . . . 6 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
27	26	eqcomi 2631	. . . . 5 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
28	22	recld2 22617	. . . . . . . . . 10 |- RR e. ( Clsd ` ( TopOpen ` ℂfld ) )
29		neg1rr 11125	. . . . . . . . . . . 12 |- -u 1 e. RR
30		iocmnfcld 22572	. . . . . . . . . . . 12 |- ( -u 1 e. RR -> ( -oo (,] -u 1 ) e. ( Clsd ` ( topGen ` ran (,) ) ) )
31	29, 30	ax-mp 5	. . . . . . . . . . 11 |- ( -oo (,] -u 1 ) e. ( Clsd ` ( topGen ` ran (,) ) )
32	22	tgioo2 22606	. . . . . . . . . . . 12 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
33	32	fveq2i 6194	. . . . . . . . . . 11 |- ( Clsd ` ( topGen ` ran (,) ) ) = ( Clsd ` ( ( TopOpen ` ℂfld ) ↾t RR ) )
34	31, 33	eleqtri 2699	. . . . . . . . . 10 |- ( -oo (,] -u 1 ) e. ( Clsd ` ( ( TopOpen ` ℂfld ) ↾t RR ) )
35		restcldr 20978	. . . . . . . . . 10 |- ( ( RR e. ( Clsd ` ( TopOpen ` ℂfld ) ) /\ ( -oo (,] -u 1 ) e. ( Clsd ` ( ( TopOpen ` ℂfld ) ↾t RR ) ) ) -> ( -oo (,] -u 1 ) e. ( Clsd ` ( TopOpen ` ℂfld ) ) )
36	28, 34, 35	mp2an 708	. . . . . . . . 9 |- ( -oo (,] -u 1 ) e. ( Clsd ` ( TopOpen ` ℂfld ) )
37		1re 10039	. . . . . . . . . . . 12 |- 1 e. RR
38		icopnfcld 22571	. . . . . . . . . . . 12 |- ( 1 e. RR -> ( 1 [,) +oo ) e. ( Clsd ` ( topGen ` ran (,) ) ) )
39	37, 38	ax-mp 5	. . . . . . . . . . 11 |- ( 1 [,) +oo ) e. ( Clsd ` ( topGen ` ran (,) ) )
40	39, 33	eleqtri 2699	. . . . . . . . . 10 |- ( 1 [,) +oo ) e. ( Clsd ` ( ( TopOpen ` ℂfld ) ↾t RR ) )
41		restcldr 20978	. . . . . . . . . 10 |- ( ( RR e. ( Clsd ` ( TopOpen ` ℂfld ) ) /\ ( 1 [,) +oo ) e. ( Clsd ` ( ( TopOpen ` ℂfld ) ↾t RR ) ) ) -> ( 1 [,) +oo ) e. ( Clsd ` ( TopOpen ` ℂfld ) ) )
42	28, 40, 41	mp2an 708	. . . . . . . . 9 |- ( 1 [,) +oo ) e. ( Clsd ` ( TopOpen ` ℂfld ) )
43		uncld 20845	. . . . . . . . 9 |- ( ( ( -oo (,] -u 1 ) e. ( Clsd ` ( TopOpen ` ℂfld ) ) /\ ( 1 [,) +oo ) e. ( Clsd ` ( TopOpen ` ℂfld ) ) ) -> ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) e. ( Clsd ` ( TopOpen ` ℂfld ) ) )
44	36, 42, 43	mp2an 708	. . . . . . . 8 |- ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) e. ( Clsd ` ( TopOpen ` ℂfld ) )
45	24	cldopn 20835	. . . . . . . 8 |- ( ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) e. ( Clsd ` ( TopOpen ` ℂfld ) ) -> ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) e. ( TopOpen ` ℂfld ) )
46	44, 45	ax-mp 5	. . . . . . 7 |- ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) e. ( TopOpen ` ℂfld )
47	3, 46	eqeltri 2697	. . . . . 6 |- D e. ( TopOpen ` ℂfld )
48	47	a1i 11	. . . . 5 |- ( T. -> D e. ( TopOpen ` ℂfld ) )
49	11, 17, 18, 20, 21, 27, 22, 48	dvmptres 23726	. . . 4 |- ( T. -> ( CC _D ( x e. D |-> ( pi / 2 ) ) ) = ( x e. D |-> 0 ) )
50	5	sseli 3599	. . . . . 6 |- ( x e. D -> x e. CC )
51		asincl 24600	. . . . . 6 |- ( x e. CC -> (arcsin ` x ) e. CC )
52	50, 51	syl 17	. . . . 5 |- ( x e. D -> (arcsin ` x ) e. CC )
53	52	adantl 482	. . . 4 |- ( ( T. /\ x e. D ) -> (arcsin ` x ) e. CC )
54		ovexd 6680	. . . 4 |- ( ( T. /\ x e. D ) -> ( 1 / ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) e. _V )
55	3	dvasin 33496	. . . . 5 |- ( CC _D (arcsin |` D ) ) = ( x e. D |-> ( 1 / ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) )
56		asinf 24599	. . . . . . . 8 |- arcsin : CC --> CC
57	56	a1i 11	. . . . . . 7 |- ( T. -> arcsin : CC --> CC )
58	57, 21	feqresmpt 6250	. . . . . 6 |- ( T. -> (arcsin |` D ) = ( x e. D |-> (arcsin ` x ) ) )
59	58	oveq2d 6666	. . . . 5 |- ( T. -> ( CC _D (arcsin |` D ) ) = ( CC _D ( x e. D |-> (arcsin ` x ) ) ) )
60	55, 59	syl5reqr 2671	. . . 4 |- ( T. -> ( CC _D ( x e. D |-> (arcsin ` x ) ) ) = ( x e. D |-> ( 1 / ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) ) )
61	11, 14, 16, 49, 53, 54, 60	dvmptsub 23730	. . 3 |- ( T. -> ( CC _D ( x e. D |-> ( ( pi / 2 ) - (arcsin ` x ) ) ) ) = ( x e. D |-> ( 0 - ( 1 / ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) ) ) )
62	61	trud 1493	. 2 |- ( CC _D ( x e. D |-> ( ( pi / 2 ) - (arcsin ` x ) ) ) ) = ( x e. D |-> ( 0 - ( 1 / ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) ) )
63		df-neg 10269	. . . 4 |- -u ( 1 / ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) = ( 0 - ( 1 / ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) )
64		1cnd 10056	. . . . 5 |- ( x e. D -> 1 e. CC )
65		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
66	50	sqcld 13006	. . . . . . 7 |- ( x e. D -> ( x ^ 2 ) e. CC )
67		subcl 10280	. . . . . . 7 |- ( ( 1 e. CC /\ ( x ^ 2 ) e. CC ) -> ( 1 - ( x ^ 2 ) ) e. CC )
68	65, 66, 67	sylancr 695	. . . . . 6 |- ( x e. D -> ( 1 - ( x ^ 2 ) ) e. CC )
69	68	sqrtcld 14176	. . . . 5 |- ( x e. D -> ( sqr ` ( 1 - ( x ^ 2 ) ) ) e. CC )
70		eldifn 3733	. . . . . . . 8 |- ( x e. ( CC \ ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) ) -> -. x e. ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) )
71	70, 3	eleq2s 2719	. . . . . . 7 |- ( x e. D -> -. x e. ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) )
72		mnfxr 10096	. . . . . . . . . . . 12 |- -oo e. RR*
73	29	rexri 10097	. . . . . . . . . . . 12 |- -u 1 e. RR*
74		mnflt 11957	. . . . . . . . . . . . 13 |- ( -u 1 e. RR -> -oo < -u 1 )
75	29, 74	ax-mp 5	. . . . . . . . . . . 12 |- -oo < -u 1
76		ubioc1 12227	. . . . . . . . . . . 12 |- ( ( -oo e. RR* /\ -u 1 e. RR* /\ -oo < -u 1 ) -> -u 1 e. ( -oo (,] -u 1 ) )
77	72, 73, 75, 76	mp3an 1424	. . . . . . . . . . 11 |- -u 1 e. ( -oo (,] -u 1 )
78		eleq1 2689	. . . . . . . . . . 11 |- ( x = -u 1 -> ( x e. ( -oo (,] -u 1 ) <-> -u 1 e. ( -oo (,] -u 1 ) ) )
79	77, 78	mpbiri 248	. . . . . . . . . 10 |- ( x = -u 1 -> x e. ( -oo (,] -u 1 ) )
80	37	rexri 10097	. . . . . . . . . . . 12 |- 1 e. RR*
81		pnfxr 10092	. . . . . . . . . . . 12 |- +oo e. RR*
82		ltpnf 11954	. . . . . . . . . . . . 13 |- ( 1 e. RR -> 1 < +oo )
83	37, 82	ax-mp 5	. . . . . . . . . . . 12 |- 1 < +oo
84		lbico1 12228	. . . . . . . . . . . 12 |- ( ( 1 e. RR* /\ +oo e. RR* /\ 1 < +oo ) -> 1 e. ( 1 [,) +oo ) )
85	80, 81, 83, 84	mp3an 1424	. . . . . . . . . . 11 |- 1 e. ( 1 [,) +oo )
86		eleq1 2689	. . . . . . . . . . 11 |- ( x = 1 -> ( x e. ( 1 [,) +oo ) <-> 1 e. ( 1 [,) +oo ) ) )
87	85, 86	mpbiri 248	. . . . . . . . . 10 |- ( x = 1 -> x e. ( 1 [,) +oo ) )
88	79, 87	orim12i 538	. . . . . . . . 9 |- ( ( x = -u 1 \/ x = 1 ) -> ( x e. ( -oo (,] -u 1 ) \/ x e. ( 1 [,) +oo ) ) )
89	88	orcoms 404	. . . . . . . 8 |- ( ( x = 1 \/ x = -u 1 ) -> ( x e. ( -oo (,] -u 1 ) \/ x e. ( 1 [,) +oo ) ) )
90		elun 3753	. . . . . . . 8 |- ( x e. ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) <-> ( x e. ( -oo (,] -u 1 ) \/ x e. ( 1 [,) +oo ) ) )
91	89, 90	sylibr 224	. . . . . . 7 |- ( ( x = 1 \/ x = -u 1 ) -> x e. ( ( -oo (,] -u 1 ) u. ( 1 [,) +oo ) ) )
92	71, 91	nsyl 135	. . . . . 6 |- ( x e. D -> -. ( x = 1 \/ x = -u 1 ) )
93		sq1 12958	. . . . . . . . . 10 |- ( 1 ^ 2 ) = 1
94		1cnd 10056	. . . . . . . . . . 11 |- ( ( x e. CC /\ ( sqr ` ( 1 - ( x ^ 2 ) ) ) = 0 ) -> 1 e. CC )
95		sqcl 12925	. . . . . . . . . . . 12 |- ( x e. CC -> ( x ^ 2 ) e. CC )
96	95	adantr 481	. . . . . . . . . . 11 |- ( ( x e. CC /\ ( sqr ` ( 1 - ( x ^ 2 ) ) ) = 0 ) -> ( x ^ 2 ) e. CC )
97	65, 95, 67	sylancr 695	. . . . . . . . . . . . 13 |- ( x e. CC -> ( 1 - ( x ^ 2 ) ) e. CC )
98	97	adantr 481	. . . . . . . . . . . 12 |- ( ( x e. CC /\ ( sqr ` ( 1 - ( x ^ 2 ) ) ) = 0 ) -> ( 1 - ( x ^ 2 ) ) e. CC )
99		simpr 477	. . . . . . . . . . . 12 |- ( ( x e. CC /\ ( sqr ` ( 1 - ( x ^ 2 ) ) ) = 0 ) -> ( sqr ` ( 1 - ( x ^ 2 ) ) ) = 0 )
100	98, 99	sqr00d 14180	. . . . . . . . . . 11 |- ( ( x e. CC /\ ( sqr ` ( 1 - ( x ^ 2 ) ) ) = 0 ) -> ( 1 - ( x ^ 2 ) ) = 0 )
101	94, 96, 100	subeq0d 10400	. . . . . . . . . 10 |- ( ( x e. CC /\ ( sqr ` ( 1 - ( x ^ 2 ) ) ) = 0 ) -> 1 = ( x ^ 2 ) )
102	93, 101	syl5req 2669	. . . . . . . . 9 |- ( ( x e. CC /\ ( sqr ` ( 1 - ( x ^ 2 ) ) ) = 0 ) -> ( x ^ 2 ) = ( 1 ^ 2 ) )
103	102	ex 450	. . . . . . . 8 |- ( x e. CC -> ( ( sqr ` ( 1 - ( x ^ 2 ) ) ) = 0 -> ( x ^ 2 ) = ( 1 ^ 2 ) ) )
104		sqeqor 12978	. . . . . . . . 9 |- ( ( x e. CC /\ 1 e. CC ) -> ( ( x ^ 2 ) = ( 1 ^ 2 ) <-> ( x = 1 \/ x = -u 1 ) ) )
105	65, 104	mpan2 707	. . . . . . . 8 |- ( x e. CC -> ( ( x ^ 2 ) = ( 1 ^ 2 ) <-> ( x = 1 \/ x = -u 1 ) ) )
106	103, 105	sylibd 229	. . . . . . 7 |- ( x e. CC -> ( ( sqr ` ( 1 - ( x ^ 2 ) ) ) = 0 -> ( x = 1 \/ x = -u 1 ) ) )
107	106	necon3bd 2808	. . . . . 6 |- ( x e. CC -> ( -. ( x = 1 \/ x = -u 1 ) -> ( sqr ` ( 1 - ( x ^ 2 ) ) ) =/= 0 ) )
108	50, 92, 107	sylc 65	. . . . 5 |- ( x e. D -> ( sqr ` ( 1 - ( x ^ 2 ) ) ) =/= 0 )
109	64, 69, 108	divnegd 10814	. . . 4 |- ( x e. D -> -u ( 1 / ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) = ( -u 1 / ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) )
110	63, 109	syl5eqr 2670	. . 3 |- ( x e. D -> ( 0 - ( 1 / ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) ) = ( -u 1 / ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) )
111	110	mpteq2ia 4740	. 2 |- ( x e. D |-> ( 0 - ( 1 / ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) ) ) = ( x e. D |-> ( -u 1 / ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) )
112	9, 62, 111	3eqtri 2648	1 |- ( CC _D (arccos |` D ) ) = ( x e. D |-> ( -u 1 / ( sqr ` ( 1 - ( x ^ 2 ) ) ) ) )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   (,)cioo 12175   (,]cioc 12176   [,)cico 12177   ^cexp 12860   sqrcsqrt 13973   picpi 14797   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746  TopOnctopon 20715   Clsdccld 20820   _D cdv 23627  arcsincasin 24589  arccoscacos 24590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-tan 14802  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304  df-asin 24592  df-acos 24593

This theorem is referenced by:  dvreacos  33499